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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On the sequence space $l_{(p_{n})}$ and $\lambda _{(p_{n})},$ $0<p_{n}\leq 1$
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by S. A. Schonefeld and W. J. Stiles PDF
Trans. Amer. Math. Soc. 225 (1977), 243-257 Request permission

Abstract:

Let $({p_n})$ and $({q_n})$ be sequences in the interval $(0,1]$, let ${l_{({p_n})}}$ be the set of all real sequences $({x_n})$ such that $\sum {|{x_n}{|^{{p_n}}} < \infty }$, and let ${\lambda _{({q_n})}}$ be the set of all real sequences $({y_n})$ such that ${\sup _\pi }\sum {|{y_n}{|^{{q_{\pi (n)}}}} < \infty }$ where the sup is taken over all permutations $\pi$ of the positive integers. The purpose of this paper is to investigate some of the properties of these spaces. Our results are primarily concerned with (1) conditions which are necessary and/or sufficient for ${l_{({p_n})}}$ (resp., ${\lambda _{({p_n})}}$) to equal ${l_{({q_n})}}$ (resp., ${\lambda _{({q_n})}}$), and (2) isomorphic and topological properties of the subspaces of these spaces. In connection with (1), we show that the following four conditions are equivalent for any sequence $({\varepsilon _n})$ which decreases to zero and has ${\varepsilon _1} < 1$. (a) There exists a number $K > 1$ such that the series $\sum {1/{K^{1/{\varepsilon _n}}}}$ converges; (b) the elements ${\varepsilon _n}$ of the sequence satisfy the condition ${\varepsilon _n} = O(1/\ln n)$; (c) the sequence $((\ln n)((1/n)\sum \nolimits _1^n {{\varepsilon _j}} ))$ is bounded; and (d) ${l_{(1 - {\varepsilon _n})}}$ equals ${l_1}$. In connection with (2), we show that the following are true when $({p_n})$ increases to one. (a) ${\lambda _{({p_n})}}$ contains an infinite-dimensional closed subspace where the ${l_{({p_n})}}$-topology and the ${\lambda _{({p_n})}}$-topology agree; (b) ${l_{({p_n})}}$ and ${\lambda _{({p_n})}}$ contain closed subspaces isomorphic to ${l_1}$; and (c) ${\lambda _{({p_n})}}$ contains no infinite-dimensional subspace where the ${\lambda _{({p_n})}}$-topology agrees with the ${l_1}$-topology if and only if \[ \lim ({(1/n)^{{p_1}}} + {(1/n)^{{p_2}}} + \cdots + {(1/n)^{{p_n}}}) = \infty .\]
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 225 (1977), 243-257
  • MSC: Primary 46A45; Secondary 46A15
  • DOI: https://doi.org/10.1090/S0002-9947-1977-0448027-X
  • MathSciNet review: 0448027